2 Answers
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On most mixers, the meter shows you the voltage swing (intensity) of the waveform. The perceived volume of sound is actually more dependent on both the intensity and frequency components of what is heard and felt. (That's what makes loudness controls and vocal exciters work.)

As far as running a high level out of the mixer, I agree with you. All of my live mixing background has been in the analog world where noise was the enemy. To that end, I always ran things as hot as possible as far as possible, while leaving a fair amount of headroom to both mix and to compensate for musicians who "give that extra effort" in performance that they don't do during the sound check.

AFAIK, on a GOOD digital system, the issue of noise is reduced and you have more headroom to play with.

The signals should in general throughout the whole system, be as high as possible without clipping (the exception on this is from preamps to poweramps, more on that later). When it comes to digital the difference is that the clipping is very hard, so it's more noticeable, so you need to be more careful (but on the other hand, you'll also notice it quicker).

There isn't any absolute connection between the meters and the perceived volume, as that entirely depends on how much volume your power amps and speakers are outputting. There is an obvious relative connection, in as much as as long as you don't change any levels on the monitoring system, higher meter levels mean higher volume levels.

The levels on the power amps should be so high that you feel it's the right level. In a live setting you can connect the power amps to the main output of the board if the amps have level adjustments as well, but it's usually handy to have an extra knob for volume adjustments, and therefore you often connect the power amps via a preamp, so you can adjust the volume separately from the main output, as you otherwise would have to adjust all the power amps separately (there are typically at least two, one left, one right, and in some cases can be many more, depending on the size of the system). This is so handy that many live mixers have dedicated speaker outputs with a separate volume for this (sometimes called "master level", each brand tends to have different names). On studio mixers it's completely essential, and there the outputs are often called "control room output" or similar.

This way you can make sure the mixer main level is nice and close to the 0db mark, while adjusting the sound level in the room either with the monitor level knob on the mixer, or the volume level on the preamp. This also means you have a good out level to stick into a system for recording the live audio, if you want to do that.